A number of theorems including weak duality theorem, duality theorem, unboundedness theorem, and existence theorem are presented in the chapter to show the relations between the primal and the dual problems. The chapter also describes how duality relations in linear programming can be used in ...
duality theorem [dü′al·əd·ē ‚thir·əm] (mathematics) A theorem which asserts that for a givenn-dimensional space, the (n-p) dimensional homology group is isomorphic to ap-dimensional cohomology group for eachp= 0, …,n, provided certain conditions are met. ...
The theorem is proved by applying a general duality theorem of Rockafellar. As applications, a dual is found for the multi-facility location problem and a nonlinear dual is obtained for a linear programming problem with a priori bounds for the variables. When the norms concerned are continuously...
Theorem 4.1 Weak duality Let λTf be (p, r) − ρ1 − (η, θ)-invex in the first variable and let −λTf be (p, − r) − ρ2 − (η, θ)-invex in the second variable. Let (x, y) and (u, v) be the feasible points for (SMP) and (SMD) respectively with ...
By Arzéla-Ascoli's theorem, we can extract a subsequence {ϕnk} of {ϕncc¯} and a subsequence {ψnk} of {ϕnc} that both converge uniformly. Let ϕnk→ϕ and ψnk→ψ . By uniform convergence, we have I(ϕnk,ψnk)→I(ϕ,ψ) and ϕ(x)+ψ(y)≤c(x,y) . ...
Oh also, you can prove that the solutions to discrete optimal transport are "sparse" i.e there always exist a solution with at most N+MN+M non zero coefficients (it's a consequence of Dubins theorem, I can give further details if you're interested). Another interesting things is the ...
Some closedness criteria for the linear image of a closed convex cone are studied in [25], and the results therein allow to obtain the necessary conditions for a conic linear system to satisfy uniform LP duality (Theorem 1 in [22]). If the cone defining this system is nice, then these...
A large chunk of the work in SVMs is converting the original, geometric problem statement, that of maximizing the margin of a linear separator, into a form suitable for this theorem. We did that last time. However, the conditions of this theorem also provide the structure for a more ...
On the Theory of Semi-Infinite Programming and a Generalization of the Kuhn-Tucker Saddle Point Theorem for Arbitrary Convex Functions We first present a survey on the theory of semi-infinite programming as a generalization of linear programming and convex duality theory. By the pairing of a finit...
In this paper we consider the duality gap functiongthat measures the difference between the optimal values of the primal problem and of the dual problem in linear programming and in linear semi-infinite programming. We analyze its behavior when the data defining these problems may be perturbed, ...